Contributions of plasma physics to chaos and nonlinear
dynamics
Dominique Escande
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Contributions of plasma physics to chaos and
nonlinear dynamics
D F Escande
Aix-Marseille Universite, CNRS, PIIM, UMR 7345, case 321, campus Saint-Jerome,
FR-13013 Marseille, France
E-mail: [email protected]
Abstract. This review focusses on the contributions of plasma physics to chaos and
nonlinear dynamics bringing new methods which are or can be used in other scientific
domains. It starts with the development of the theory of Hamiltonian chaos, and then
deals with order or quasi order, for instance adiabatic and soliton theories. It ends
with a shorter account of dissipative and high dimensional Hamiltonian dynamics,
and of quantum chaos. Most of these contributions are a spin-off of the research on
thermonuclear fusion by magnetic confinement, which started in the fifties. Their
presentation is both exhaustive and compact. [21 July 2016]
PACS numbers : 01.65.+g, 05.45.-a, 52.
Keywords : history, plasma physics, nonlinear dynamics, classical chaos, quantum chaos
Contributions of plasma physics to chaos and nonlinear dynamics 2
1. Introduction
In the second half of the twentieth century, the magnetic confinement of charged particles
in magnetic bottles was studied to heat plasmas to the high temperatures required
for controlled thermonuclear fusion. The basic idea was to put particles in magnetic
fields whose lines wind upon magnetic surfaces. Rapidly, it turned out that these
lines are described by low dimensional Hamiltonian mechanics, forcing physicists to
revisit classical mechanics and to get acquainted with the related results mathematicians
started to develop in the first half of the twentieth century†. Unfortunately, though very
useful, these results were not sufficient to deal with magnetic confinement. Fortunately,
the development of computers and of numerical calculations enabled them to visualize
dynamics, to compute their features, and to back up the development of non rigorous
approaches. While textbooks were full of examples with regular motion, chaos turned
out to be an ubiquitous behaviour of classical mechanics!
A parallel endeavour in the second half of the twentieth century occurred in the
context of accelerator physics, with the aim to drive particles to the high energies
necessary for studying the physics of elementary particles. This means avoiding chaos
and not describing it, which limits the scope of studies on this topic. In contrast, the
quest for magnetic fusion requires also the study of chaos itself, in order to understand
chaotic transport. Indeed, this issue is present in very different parts of the research
on magnetic fusion: magnetic field line topology, dynamics of particles in magnetic
fields, turbulent transport, radiofrequency heating, ray dynamics, etc‡... This broad
scope of problems naturally triggered a series of contributions to nonlinear dynamics
and chaos. These contributions are the topic of the present topical review. In order
to limit its length, it focusses on the contributions bringing new methods which are or
can be used in other scientific domains. These contributions are of several types. There
are those of plasma physicists and those inspired by plasma physics to scientists of
other fields of science, mathematicians and astronomers in particular. There are also
those of plasma physicists initially working on chaos and nonlinear dynamics, who then
decided to dedicate their later research to this exciting and challenging topic. Plasma
physics may not vindicate their whole successive work, but at least the initial impetus
it provided them.
This topical review starts with the development of the theory of Hamiltonian chaos
in the context of plasmas (section 2). Indeed, this was a turning point in plasma
physics, because this development uncovered the then mysterious dynamics underlying
phenomena traditionally tackled by statistical approaches; in particular, thanks to
images provided by numerical calculations. This induced a strong interest in the whole
plasma community, even by non practitioners of chaos.
† (Morrison 2000b) makes at length the point that plasma physics re-ignited research in classical
dynamics.‡ Chaos is at work in particular in anomalous transport of fusion plasmas, when heating magnetized
plasmas by cyclotronic waves, and in the deep penetration of lower hybrid rays in such plasmas.
Contributions of plasma physics to chaos and nonlinear dynamics 3
The dynamics of complex systems generally exhibits a mixture of ordered and
chaotic motion. Furthermore, a chaotic orbit needs a sufficient time to really look
chaotic. Therefore, there is a large part of the nonlinear dynamics of plasmas which
deals with order or quasi order, for instance adiabatic theory. This is the topic of the
second part of this topical review (section 3). It deals, in particular, with soliton theory,
which is an early landmark of the contributions of plasma physics to nonlinear dynamics.
The following parts are shorter and deal with high dimensional Hamiltonian
dynamics (section 4), dissipative dynamics (section 5), quantum chaos (section 6), and
possible extensions (section 7).
Writing the history of the contributions of plasma physics to chaos and nonlinear
dynamics is a delicate endeavour. Indeed, the beginning of this history corresponds
to a literature whose access is often not easy, in particular for linguistic reasons, while
the more modern part suffers from the present deluge of publications where interesting
results may be drowned. In order to limit bias, a series of colleagues listed in the
acknowledgements kindly provided me with their views on the topic of this topical
review. This considerably enriched its initial scope, but it may still be incomplete.
Telling the same story with equations and figures would require a book. The present
compact and hopefully close to exhaustive paper is tailored for the plasma physics
community, and especially the fusion one, where most concepts underlying this story
are well-known. In any event, more details can be obtained in the about 250 quoted
references with a simple click. However, in order to help non specialists to access more
easily to the basics of the main issues, references to sections of review papers and
textbooks are provided in the corresponding sections of this review and are indicated
with “Pedagogy”.
2. Hamiltonian chaos
2.1. How did the story start?
After the second world war, classical mechanics was not fashionable at all among
physicists. Why did plasma physics happen to contribute to this field? It is because
many plasma physicists were involved in the development of thermonuclear fusion by
magnetic confinement. The story can be told as follows.
In 1950, Spitzer invents the stellarator, and in 1951 Sakharov and Tamm invent
the tokamak. While the principle of the latter comes with a regular magnetic field, this
cannot be taken for granted for the former, a serious issue for a magnetic configuration
meant to be a magnetic bottle! This forces theoreticians to start the still on-going
study of the nature of magnetic field lines of stellarators. Even if a stellarator looks
like a figure 8 as Spitzer’s one, topologically it is a torus. Therefore, the regularity of
its magnetic field lines can be checked by looking at their successive intersections with
the surface of section defined by a given toroidal angle. A massless charged particle
streaming freely along a given magnetic field line crosses such a surface periodically in
Contributions of plasma physics to chaos and nonlinear dynamics 4
time. By analogy, it is natural to consider this line as an orbit parameterized by a
time which is the toroidal angle. Then, studying the nature of magnetic field lines boils
down to the study of the nature of orbits in a torus, and their successive intersections
build the so-called Poincare map of their dynamics†. Magnetic flux conservation makes
such a map area-preserving, implying a relationship between magnetic field lines and
Hamiltonian systems.
While in 1952 Kruskal‡ introduces and iterates area preserving maps for stellarator
magnetic field lines (Kruskal 1952), they are fully recognized as Hamiltonian systems
only ten years later by fusion physicists (Kerst 1962, Gelfand, Graev, Zueva, Morozov
& Solov’ev 1962, Morozov, Solov’ev, & Leontovich 1966). However, the general and
explicit Hamiltonian description for field lines is provided by Boozer even two decades
after (Boozer 1981)§. Two years later, a more fundamental description is given by
two other plasma physicists, Cary and Littlejohn, in terms of an action principle that
depends on the vector potential‖ (Cary & Littlejohn 1983)¶. In 1986, Elsasser completes
the picture by showing the equivalence of changes of gauge and of canonical variables
(Elsasser 1986).
Having a Hamiltonian description of magnetic field lines is nice, but when coming
to the numerical calculation of Poincare maps, the integration of orbits from differential
equations is a formidable task for the computers of the sixties! This motivates physicists
to derive explicit area preserving maps corresponding to a full step of the Poincare map.
The paradigm of such maps is the standard map, also called Chirikov-Taylor map.
This map appeared first in 1960 in the context of electron dynamics in the microtron+
(Kolomenskii 1960), a type of particle accelerator concept originating from the cyclotron
in which the accelerating field is not applied through large D-shaped electrodes, but
through a linear accelerator structure. This map was independently proposed and
† This technique is also used experimentally. The first instance is the mapping of the magnetic field
lines of the model-C Stellarator (Sinclair, Hosea & Sheffield 1970).‡ Kruskal is quoted several time in this topical review. Indeed, he made essential contributions to
nonlinear dynamics and chaos. He was also quite influential. In particular in astrophysics, as can be
seen in the acknowledgements of the Henon-Heiles paper where he is thanked (Henon & Heiles 1964).
Indeed, this famous work was performed while Michel Henon was in Princeton. It describes the non-
linear motion of a star around a galactic center where the motion is restricted to a plane, and uncovers
this motion can be chaotic.§ Boozer is a plasma physicist whose paper is in Physics of Fluids which published both fluid and
plasma papers at that time.‖ Reference (Pina & Ortiz 1988) describes how to implement this technique in concrete cases and
illustrates its large flexibility.¶ This action principle is present implicitely in (Morozov et al. 1966) where a zero gyro-radius limit of
the particle action was taken.+ According to reference 2 of (Melekhin 1975), it appeared ten years earlier in Kolomenskii’s PhD
thesis at the Lebedev Institute.
Contributions of plasma physics to chaos and nonlinear dynamics 5
numerically studied in a magnetic fusion context by Taylor † in 1968, and by Chirikov‡
in 1969 (also for particle accelerators) (Chirikov 1969). The latter also recovers this map
on linearizing the motion in the vicinity of a separatrix (section 4.4 of (Chirikov 1979)).
2.2. Transition to chaos in Hamiltonian systems
In the sixties, numerical simulations make visible that the phase-space of a typical
1.5 degree-of-freedom Hamiltonian system has intertwined zones of regular and chaotic
orbits, a fact known by mathematicians for decades. For plasma physicists it is
important to know how broad are chaotic domains§, and the threshold where such
domains connect to make sizeable chaotic “seas” for systems with a control parameter.
The latter issue leads Chirikov to derive in 1959 an approximate criterion (Chirikov 1959,
Chirikov 1979), described below, whose use becomes rapidly ubiquitous. This criterion
works for Hamiltonians, which are the sum of an integrable part and of a perturbation
written in terms of the action-angle variables of the integrable part. Kolmogorov-Arnold-
Moser (KAM) theory (Kolmogorov 1954, Moser 1962, Arnold 1963a) reveals that the
extension of chaotic domains is bounded by KAM tori‖. Therefore, a more rigorous
approach to the estimate of the width of chaotic domains goes through the estimate of
the threshold of break-up of KAM tori. Greene¶ achieves this in 1979 in a very accurate
way through a criterion (Greene 1979), described below, which is explicitly implemented
for the standard map+. As a result, the transition to chaos is then described in two
complementary ways using Hamiltonians and maps. The next two subsections tell the
corresponding story. The first one describes the Hamiltonian approach which started
first historically.
† Taylor writes : “At the time I was interested in [the standard map], I had just obtained an early desk
calculator which could be programmed using three stored addresses (x, y, z), but more importantly it
could be connected to a mechanical plotter! One lunch time I set it up to iterate the equations above,
and to plot the successive points - which it did at about two per second! When I returned from lunch
there was the first picture of regular and chaotic regions. The others followed later. I did not consider
this work suitable for publication, but I did include the figures in the Culham Progress report for
that year (1968-9). This was noticed and taken up by others, notably Froeschle (Froeschle 1970) who
published the figures (with acknowledgment) and by Stix, who included them in his lectures. So the
model became quite well known.” (Taylor 2015).‡ Chirikov is quoted repeatedly in this paper for contributions in many different problems of nonlinear
dynamics and chaos. A summary of his main contributions can be found in (Bellissard, Bohigas, Casati
& Shepelyansky 1999), published in a special issue of Physica D in his honor.§ Then called “stochastic domains”. The change of name is motivated by several reasons. A simple
one comes from the inspection of the time evolution of two nearby orbits at initial time. In a chaotic
system they diverge exponentially, but not in a stochastic one.‖ The quotations of (Arnold 1963b) and (Moser 1968) show that plasma physics is also a source of
inspiration for these mathematicians.¶ Greene is quoted repeatedly in this paper for contributions in many different problems of nonlinear
dynamics and chaos. A summary of his main contributions can be found in (Morrison, Johnson &
Chan 2008).+ Scalings in (Greene 1968) anticipate this result.
Contributions of plasma physics to chaos and nonlinear dynamics 6
2.2.1. Working with Hamiltonians
Resonance-overlap criterion In 1959, Chirikov deals with the confinement of charged
particles in magnetic mirror traps. He focusses on resonances between the Larmor
rotation of charged particles and their slow oscillations along the lines of force.
In agreement with numerical simulations, he hypothesizes that when neighboring
resonances overlap, there is a complete exchange of energy among the degrees of
freedom of the particle, so that the particle escapes from the trap (Chirikov 1959)
(Pedagogy: section 4.2 of (Lichtenberg & Lieberman 2013) and section 5.2.2 of (Elskens
& Escande 2003)); in modern language, this corresponds to a large scale chaotic motion
of the particle. This criterion is immediately applied successfully to the determination
of the confinement threshold for experiments with plasmas of open mirror traps†
(Rodionov 1959). This simple to implement criterion becomes rapidly famous among
physicists‡, and especially after Chirikov’s review paper§ (Chirikov 1979). The resonant
domain of a single wave and the overlap of the resonance domains of two waves can
be observed in a traveling wave tube, a kind of noiseless beam-plasma system (Doveil,
Auhmani, Macor & Guyomarc’h 2005). Rechester and Stix, when dealing with magnetic
chaos due to weak asymmetry in a tokamak, use this criterion to compute the width
of narrow chaotic (“stochastic”) layers next to the separatrix of an integrable system
when it is perturbed‖ (Rechester & Stix 1976).
This criterion is useful for systems with many degrees of freedom too. Indeed, it
can be directly applied to determine the energy border for strong chaos in the Fermi-
Pasta-Ulam system when only a few long wave modes are initially excited (Chirikov
& Izrailev 1966, Chirikov, Izrailev & Tayursky 1973). In (Escande, Kantz, Livi &
Ruffo 1994) one computes the Gibbsian probability distribution of the overlap parameter
s corresponding to two nearby resonances of the Hamiltonian of a chain of rotators.
Requiring the support of this distribution to be above the threshold of large scale chaos,
gives the right threshold in energy above which the Gibbsian estimate of the specific heat
† References (Chirikov 1987, Chirikov 1987b, Chirikov 1993) yield a broad account of particle
confinement in magnetic traps.‡ An indication of the importance of this criterion is obtained when typing “resonance overlap
criterion” in Google Scholar: 209,000 references are obtained; (Chirikov 1979) is quoted 4,200 times
according to the Web of Science. Though not quoting Chirikov’s seminal work, reference (Rosenbluth,
Sagdeev & Taylor 1966) contributed to publicize the concept of resonance overlap too. In reality,
the resonances supposed to overlap exist only in the limit where only one is present and the system
is integrable. When the size of both trapping domains increases, a Poincare map shows that chaos
makes progressively the integrable separatrices fuzzy, which rules out any overlap. Furthermore, a
perturbative calculation shows there is a mutual repulsion of the separatrices, which makes their overlap
more difficult (Escande 1979). Therefore, the resonance overlap criterion should be more adequately
named a “heteroclinic connection criterion” (see figure 1 of (Elskens & Escande 1993)).§ This paper brings also a wealth of information about Hamiltonian chaos which is very useful for
physicists of the eighties to get acquainted with chaos theory (“stochasticity theory” at that time).‖ Here again, Chirikov’s authorship of the resonance overlap criterion is overlooked. First estimates of
the width of chaotic layers are given in (Zaslavsky & Filonenko 1968, Zaslavsky & Chirikov 1971) with
the overlap criterion. Rechester and Stix’ estimates are improved in (Escande 1982a).
Contributions of plasma physics to chaos and nonlinear dynamics 7
at constant volume agrees with the time average of the estimate given by the fluctuations
of the kinetic energy: one has a self-consistent check of the validity of Gibbs calculus
using the observable s!
Renormalization approach At the end of the seventies, Doveil is working on ion acoustic
waves in a multipole plasma device where these waves are dispersive. This leads him
to study numerically the (chaotic) dynamics of ions when several waves are present†.
For simplicity he focusses on the two-wave case. The Poincare map corresponding to
the motion of one particle in two longitudinal (plasma) waves displays a sequence of
resonance islands related to higher order nonlinear resonances which become explicit
by canonical transformations‡. Following the philosophy of Chirikov’s review paper
(Chirikov 1979), it is then tempting to apply the resonance overlap criterion to two
neighboring such resonances. The criterion is easier to apply if the corresponding
Hamiltonian is approximated by a simpler one... corresponding to the motion of one
particle in two longitudinal waves. The passage from the initial two-wave Hamiltonian to
the transformed one is similar to the transform of Kadanoff’s block-spin renormalization
group (Escande & Doveil 1981a, Escande & Doveil 1981b) (Pedagogy: section 4.5
of (Lichtenberg & Lieberman 2013) and section 5.4 of (Elskens & Escande 2003))...
where Chirikov’s criterion is absent! Its practical implementation for more general
Hamiltonians is then described in (Escande, Mohamed-Benkadda & Doveil. 1984) and
in sections 3.1 and 4.1 of the review paper (Escande 1985). Appendix B of the latter
reference shows how to derive a renormalization for any KAM torus trapped into a
resonance island. A one parameter renormalization scheme is derived for “stochastic
layers” in (Escande 1982a). All these schemes are approximate ones in a physicist
sense: the approximations are not mathematically controlled.
Later on several mathematical works try and cope with this shortcoming. A way
to make the 1981 renormalization scheme rigorous is indicated in (MacKay 1995). The
ideas proposed originally in (Escande & Doveil 1981a, Escande & Doveil 1981b) lead
to approximate renormalization transformations enabling a very precise determination
of the threshold of break-up of invariant tori for Hamiltonian systems with two
degrees of freedom (Chandre & Jauslin 2002); these transformations are similar to
the transformation of Kadanoff’s block-spin renormalization, in the sense that they
combine a process of elimination and rescaling. In (Chandre & Jauslin 2002) the break-
up of invariant tori proves to be a universal mechanism and the renormalization flow is
† Computers at that time are slow enough for the successive points of the Poincare map to come
successively on the screen of a monitor. This makes obvious that chaotic motion is not really stochastic.
In particular, when islands are present in the chaotic sea, one can see long phases where the orbit looks
as trapped in the corresponding resonances, a feature incompatible with a stochastic process. This is
due to a self-similar structure producing strong correlations and incomplete chaos (Zaslavsky, Stevens
& Weitzner 1993).‡ This set of islands has a signature, a “devil’s staircase”, which is experimentally observable in a
traveling wave tube (Macor, Doveil & Elkens 2005). The cancellation of a set of islands can build a
barrier to transport in the same device (Chandre, Ciraolo, Doveil, Lima, Macor & Vittot 2005).
Contributions of plasma physics to chaos and nonlinear dynamics 8
precisely described. In 2004, Koch brings this type of approach to a complete rigorous
proof (Koch 2004): it is a computer assisted proof of the existence of a fixed point with
non-trivial scaling for the break-up of golden mean KAM tori.
Other approaches For Hamiltonians with zero or one primary resonance, one cannot
apply the resonance overlap criterion or the above renormalization approach. Reference
(Codaccioni, Doveil & Escande 1982) shows how to compute the threshold of large scale
chaos by using the blow-up of the width of chaotic layers† as computed in (Rechester &
Stix 1976).
In 2000, the study of the Poincare map of magnetic field lines in a toroidal
confinement configuration called “reversed field pinch” germane to the tokamak is the
occasion of a paradoxical discovery: chaos decreases when a magnetic perturbation
increases, in contradiction with the prediction of the resonance overlap criterion! This
phenomenon stems from a separatrix disappearance due to an inverse saddle-node
bifurcation (Escande, Paccagnella, Cappello, Marchetto & D’Angelo 2000), a process
likely to occur in many Hamiltonian systems‡.
2.2.2. Working with maps In the seventies, Greene is interested in the nature of
magnetic field lines in stellarators, and in their corresponding return area-preserving
map. He naturally focusses on the simplest example of such maps, the standard map,
which is very easy to iterate on computers of that time. At the end of the 70’s,
in the same way as it is natural to focus on higher order nonlinear resonances in a
Hamiltonian description, it is natural to focus on periodic orbits with a long period
in area-preserving maps. These periodic orbits correspond to O-points and X-points of
resonance islands. While studying the stable periodic orbits approximating a given KAM
torus when truncating the continuous fraction expansion of its winding number at high
order, Greene notes they become unstable when the KAM torus breaks up. This leads
him to his famous “residue criterion” which provides a method for calculating, to very
high accuracy, the parameter value for the destruction of the last torus§ (Greene 1979)
(Pedagogy: section 3.2a of (Lichtenberg & Lieberman 2013)). Defining the threshold of
large scale chaos in a given domain of phase space means finding the threshold of break-
up of the most robust KAM torus. The continued fraction expansion of their winding
† One of the considered cases is the polynomial Henon-Heiles Hamiltonian (Henon & Heiles 1964).
However, the technique of (Codaccioni et al. 1982) is unable to detect integrability. Indeed, it predicts
a blow-up of the width of a chaotic layer also for the integrable Hamiltonian obtained from Henon-Heiles’
one by changing a sign in its formula!‡ It occurred in the discharges at high current of the RFX-mod reversed field pinch, first when
stimulated by a modulated edge toroidal field (Lorenzini, Terranova, Alfier, Innocente, Martines,
Pasqualotto & Zanca 2008), and then spontaneously (Lorenzini, Martines, Piovesan, Terranova, Zanca,
Zuin, Alfier, Bonfiglio, Bonomo, Canton et al. 2009).§ Four theorems show that large parts of this criterion have a firm foundation, but not all cases have
been analyzed: for instance, can a non-smooth circle have residues going to infinity (MacKay 1992)?
If so, then one cannot infer from residues going to infinity that there is not a circle.
Contributions of plasma physics to chaos and nonlinear dynamics 9
number is found numerically to have a special form exhibited in (Greene, MacKay &
Stark 1986)†.
Greene’s work is placed in a renormalization group setting by MacKay, then his
student (MacKay 1983). This work is closely related to the approximate renormalization
described above. This triggers a dialog between the latter renormalization and the
rigorous one under the auspices of Greene’s criterion during almost a decade (Schmidt
1980, Doveil & Escande 1981, Doveil & Escande 1982, Schmidt & Bialek 1982, Escande
& Mehr 1982c, Mehr & Escande 1984, MacKay 1988a). The existence of a fixed point
with a non-trivial scaling for MacKay’s renormalization is finally proved in 2010 (Arioli
& Koch 2010).
Reference (MacKay 1989) derives a simple criterion for non-existence of invariant
tori. When applied to the Hamiltonian describing the motion of a particle in the field
of two waves of section 2.2.1, it gives results in close agreement with those of Greene’s
residue method.
Till now we considered maps such that the period of the motion on a torus
is a monotonic function of the action of the torus: they are twist maps. We
now deal with non-twist systems where the period goes through an extremum on a
given torus; in such systems, the overlap criterion fails and KAM theorem cannot
be applied. They are introduced in 1984 by Howard, motivated by multifrequency
electron-cyclotron-resonance heating in plasmas (Howard & Hohs 1984). He derives an
accurate analytic reconnection threshold of the approximate separatrices of the pairs of
islands corresponding to actions symmetrical with respect to that of the extremum
period. Motivated by the study of magnetic chaos in systems with reversed shear
configurations, del-Castillo-Negrete and Morrison propose a prototype map called the
standard non-twist map, and present a detailed renormalization group study of the non-
twist transition to chaos (del Castillo-Negrete & Morrison 1992a, del Castillo-Negrete,
Greene & Morrison 1996, del Castillo-Negrete, Greene & Morrison 1997) : there is a new
universality class in this transition. This stimulates a series of rigorous mathematical
results: in (Delshams & la Llave 2000), the proof of persistence of critical circles
and a partial justification of Greene’s criterion, as generalized in (del Castillo-Negrete
et al. 1996). In (Gonzalez-Enriquez, Haro & la Llave 2014), the bifurcations of KAM
tori are studied by using the classification of critical points of a potential as provided by
Singularity Theory. This approach is applicable to both the close-to-integrable case and
the far from integrable case whenever a bifurcation of invariant tori has been detected
numerically, but the system is not necessarily written as a perturbation of an integrable
one.
2.2.3. Finite time mappings for Hamiltonian flows In view of the many techniques
which can be used for area preserving maps, it is interesting to construct a finite
† Greene’s criterion originates from stellarator studies. In 1984 it feedbacks on them by enabling the
derivation of Hamiltonian systems that are for all practical purposes integrable, with the particular
application to design stellarators with almost non-chaotic magnetic fields (Hanson & Cary 1984).
Contributions of plasma physics to chaos and nonlinear dynamics 10
time mapping corresponding to a given Hamiltonian flow. In the nineties, motivated
by various plasma physics issues, Abdullaev develops to this end his mathematically
rigorous “mapping method” based on Hamilton-Jacobi theory and classical perturbation
theory which works for Hamiltonians that are the sum of an integrable part and of a small
perturbation (Abdullaev 1999, Abdullaev 2002) (See (Abdullaev 2006, Abdullaev 2014a)
for a systematic description of the method). This method can be used to find infinitely
many periodic orbits of a perturbed Hamiltonian system, and not only primary ones
(see Sec. 7.2.2 in (Abdullaev 2014a)). It can also be used as a method of symplectic
integration, with an accuracy controlled by the product of the perturbation parameter
and of the mapping time step (Abdullaev 2002). The method enables to derive the
canonical versions of mappings widely used in the theory of chaotic Hamiltonian systems
and of their applications (see (Abdullaev 2004a, Abdullaev 2006, Abdullaev 2007) and
references therein).
Using this method, the canonical separatrix mapping describing the dynamics
near a separatrix is derived in (Abdullaev 2004b, Abdullaev 2005), while keeping the
canonical variables of the corresponding Hamiltonian, an improvement over Chirikov’s
separatrix mapping (Chirikov 1979). The new mapping is consistent with the rescaling
invariance described in section 2.3.4, and enables the derivation of analytical formulas
for the stable and unstable manifolds of the saddle point (Abdullaev 2014b, Abdullaev
2014a).
2.3. Chaotic transport
In Hamiltonian systems with 1.5 or 2 degrees-of-freedom, when KAM tori break up in
a given domain of phase space, chaotic transport sets in. In the sixties, the lack of
mathematical results enabling the description of this transport and its quantification,
leads plasma physicists to assume that chaotic motion is a stochastic one close to a
Brownian motion and has a diffusive nature. However, as shown below, this is not the
whole story!
2.3.1. Quasilinear diffusion
As an Ansatz In 1966, Rosenbluth, Sagdeev, and Taylor are interested in the
destruction of magnetic field surfaces by magnetic field irregularities (Rosenbluth
et al. 1966). They state that if there is resonance overlap, “then a Brownian motion of
flux lines and rapid destruction of surfaces results”. They go to the action-angle variables
of the unperturbed magnetic field, and write a Liouville equation for field lines which is
very similar to the Vlasov equation. Since the magnetic modes play a role for field lines
analogous to that of Langmuir waves for electrons in a plasma, they use the quasilinear
estimate of transport corresponding to the latter case. This estimate, introduced in
1961 (Romanov & Filippov 1961), had been made popular in 1962 by two papers
on the bump-on-tail instability published in two successive issues of Nuclear Fusion
Contributions of plasma physics to chaos and nonlinear dynamics 11
(Vedenov, Velikhov & Sagdeev 1962, Drummond & Pines 1962) (Pedagogy: section 5.4
of (Lichtenberg & Lieberman 2013) and section 6.8.1 of (Elskens & Escande 2003)).
Then, quasilinear estimates are made systematically for chaotic transport for almost
three decades without questioning its validity, except for the standard map which
exhibits a diffusive behavior with a diffusion constant oscillating as a function of the
control parameter of the map about the quasilinear value† (Chirikov 1979, Rechester
& White 1980, Rechester, Rosenbluth & White 1981, Meiss, Cary, Grebogi, Crawford,
Kaufman & Abarbanel 1983). However, in 1998 the diffusion properties of the Chirikov-
Taylor standard map are shown to be nonuniversal in the framework of the wave-particle
interaction, because this map corresponds to a spectrum of waves whose initial phases
are all correlated (Benisti & Escande 1998b). For the sawtooth map, the dynamics is
found diffusive for most of the integer values of the perturbation parameter (Cary &
Meiss 1981b). This is done by calculating the characteristic functions, and the joint
probabilities of the map (Cary, Meiss & Bhattacharjee 1981c).
Quasilinear or not? For a chaotic motion, the perturbative approach used in the
original derivation of the quasilinear equations cannot be justified. Therefore, the
quasilinear description might not be correct to describe the saturation of the bump-
on-tail instability; its inconsistency is shown in (Laval & Pesme 1983a, Laval & Pesme
1983b). In 1984, Laval and Pesme propose a new Ansatz to substitute the quasilinear
one, and predict that the velocity diffusion coefficient should be renormalized by a
factor 2.2 during the saturation of the instability (Laval & Pesme 1984). This motivates
Tsunoda, Doveil, and Malmberg to perform an experiment with an electron beam in
a traveling wave tube, in order to avoid the noise present in beam-plasma systems
(Tsunoda, Doveil & Malmberg 1987a, Tsunoda, Doveil & Malmberg 1987b, Tsunoda,
Doveil & Malmberg 1991). This experiment comes with a surprising result: quasilinear
predictions look right, while quasilinear assumptions are proved to be completely wrong.
Indeed no renormalization is measured, but mode-mode coupling is not negligible at all.
This sets the issue: is there a rigorous way to justify quasilinear estimates for chaotic
dynamics?
This issue is first tackled by the author of this paper by considering the self-
consistent motion of a finite number of waves and particles corresponding to the beam-
plasma problem (see the first paragraph of section 4.1). However, the motion of a particle
in a prescribed spectrum of waves is mysterious too and deserves a thorough study. For
uncorrelated phases, it is natural to expect the diffusion coefficient to converge to its
quasilinear estimate from below when the resonance overlap of the waves increases.
In 1990 numerical simulations of the motion of one particle in a spectrum of waves
performed by Verga come with a surprising result: for intermediate values of the overlap,
the diffusion coefficient exceeds its quasilinear value by a factor about 2.5 (Cary, Escande
† If the orbits inside accelerating islands are also taken into account, one finds a superdiffusive transport
(Benkadda, Kassibrakis, White & Zaslavsky 1997). Transport becomes diffusive by adding some noise
to the mapping (Karney, Rechester & White 1982).
Contributions of plasma physics to chaos and nonlinear dynamics 12
& Verga 1990).
Diffusion or not? This result triggers a series of works aiming at understanding whether
the diffusion picture makes sense and when the quasilinear estimate is correct. In 1997,
Benisti shows numerically that the diffusion picture is right, provided adequate averages
are performed on the dynamics (Benisti & Escande 1997) (see also section 6.2 of (Elskens
& Escande 2003), and (Escande & Sattin 2007, Escande & Sattin 2008)); however, this
picture is wrong if one averages only over the initial positions of particles with the same
initial velocity, in contrast with the 1966 Brownian Ansatz: chaotic does not mean
Brownian! This motivates Elskens to look for mathematical proofs using probabilistic
techniques, and leads him to two results: individual diffusion and particle decorrelation
are proved for the dynamics of a particle in a set of waves with the same wavenumber and
integer frequencies if their electric field is gaussian (Elskens & Pardoux 2010), or if their
phases have enough randomness (Elskens 2012). The intuitive reason for the validity of
the diffusive picture is given in (Benisti & Escande 1997): it is due to the locality in
velocity of the wave-particle interaction, which makes the particle to be acted upon by a
series of uncorrelated dynamics when experiencing large scale chaos. This locality of the
wave-particle interaction is rigorously proved by Benisti in (Benisti & Escande 1998a).
On taking into account that the effect of two phases on the dynamics is felt only after a
long time when there is strong resonance overlap, it can be approximately proved that
the diffusion coefficient is larger than quasilinear, but converges to this value when the
resonance overlap goes to infinity (Benisti & Escande 1997, Escande & Elskens 2002b)
(see also (Escande & Elskens 2002a, Escande & Elskens 2003), and Pedagogy: section
6.8.2 of (Elskens & Escande 2003)).
For the advection of particles in drift waves or in 2-dimensional turbulence, care
must be exerted when trying to define a corresponding diffusive transport. Then one
must define the Kubo number K which is the ratio of the correlation time of the
stochastic potential as seen by the moving object to the (nonlinear) time where the
dynamics is strongly perturbed by this potential (trapping time, chaos separation time,
Lyapunov time, ...) (Ottaviani 1992, Vlad, Spineanu, Misguich, Reuss, Balescu, Itoh
& Itoh 2004). At a given time, the potential has troughs and peaks. If the potential
is frozen, particles bounce in these troughs and peaks. When K ≪ 1, the particles
typically run only along a small arc of the trapped orbits of the instantaneous potential
during a correlation time (see figure 1a of (Escande & Sattin 2007)). During the next
correlation time they perform a similar motion in a potential completely uncorrelated
with the previous one. These uncorrelated random steps yield a 2 dimensional Brownian
motion with a diffusion coefficient which can be computed with a quasilinear estimate.
2.3.2. Diffusion with trajectory trapping If K ≫ 1, a quasi-adiabatic picture works:
the particles make a lot of bounces before the potential changes its topography (see
figure 1b of (Escande & Sattin 2007)). The change of topography forces particles to
jump to a nearby trough or peak. The successive jumps produce a random walk whose
Contributions of plasma physics to chaos and nonlinear dynamics 13
order of magnitude of the corresponding diffusion coefficient can be easily computed
(Ottaviani 1992, Vlad et al. 2004, Escande & Sattin 2007, Escande & Sattin 2008).
In a series of works, Vlad and coworkers clarify the issue of diffusion with trajectory
trapping. The just described simple picture is almost correct for a Gaussian spatial
correlation function of the potential (Vlad et al. 2004). More generally, the frozen
potential displays as well “roads” crossing the whole chaotic domain. This enables
long flights in the dynamics that bring some dependence upon K in the estimate for
the diffusion coefficient. The correct calculation of this coefficient is a much harder
task. To this end, one may group together the trajectories with a high degree of
similarity, and one starts the averaging procedure over these groups. This yields the
decorrelation trajectory method (Vlad, Spineanu, Misguich & Balescu 1998) and the
nested subensemble approach (Vlad et al. 2004). These techniques are extensively
used for the study of the transport in magnetically confined plasmas (see (Vlad &
Spineanu 2013) for a recent set of references), and for the study of astrophysical
plasmas (Vlad & Spineanu 2014) and of fluids (Vlad & Spineanu 2015). Reference (Vlad
et al. 1998) computes numerically the diffusion coefficient of particles in a spectum of
waves scaling like k−3, and for Kubo numbers up to 2105. For largeK’s, due to trajectory
trapping, the scaling of the diffusion coefficient with K is less than K1 corresponding
to the Bohm scaling. For 1 < K < 104, the results agree with the percolation scaling
K0.7 (Isichenko 1992), but the scaling K0.64 provided by the decorrelation trajectory
method fits better the data in the whole domain 1 < K < 2105. A diffusion coefficient
proportional to K2/3, better than the percolation scaling, is obtained by a simple
random-walk model using the concept of Hamiltonian pseudochaos, i.e. random non-
chaotic dynamics with zero Lyapunov exponents (Milovanov 2009).
Some turbulent plasmas may be modeled by integrable Hamiltonian systems
subjected to non-smooth perturbations. Then, chaotic transport occurs at any small
magnitude of perturbation. The profile of the diffusion coefficient in the unperturbed
action is found to have a fractal–like structure with a reduced or vanishing value of the
coefficient near low-order rational tori (Abdullaev 2011).
2.3.3. Pinch velocity In reality, when the diffusive picture is correct, there is a pinch or
dynamic friction part on top of the diffusive part, and the correct model is the Fokker-
Planck equation (Escande & Sattin 2007, Escande & Sattin 2008). For the advection of
particles in drift waves or in 2-dimensional turbulence, the direction of this pinch part
depends on K (Vlad, M. & Benkadda 2006).
2.3.4. Rescaling invariance of chaotic transport in chaotic layers Consider a one-
dimensional Hamiltonian which is the sum of an integrable part displaying a hyperbolic
fixed point X and of a time-periodic perturbation with amplitude ǫ. Its phase-space
near X turns out to be invariant with respect to a rescaling of the conjugate coordinates
along the eigenvectors of X , of ǫ, and of the phase of the perturbation. In the
middle of the nineties, Abdullaev and Zaslavsky show it numerically (Abdullaev &
Contributions of plasma physics to chaos and nonlinear dynamics 14
Zaslavsky 1994, Zaslavsky & Abdullaev 1995), and prove it rigorously (Abdullaev &
Zaslavsky 1995, Abdullaev 1997) (see also (Abdullaev 2000, Abdullaev 2006)). Since
the motion slows down near a saddle point, a particle spends relatively large time
intervals there. Therefore, the transport of particles in a narrow stochastic layer
about the separatrix related to X is mainly determined by the structure of this layer
near this point. It turns out that the statistical properties of chaotic transport are
periodic (or quasiperiodic) functions of log ǫ (Abdullaev 2000) (see also (Abdullaev &
Spatschek 1999, Abdullaev 2006)).
2.3.5. Transport through cantori In the eighties it becomes clear among plasma
physicists that chaotic transport is intrinsically more intricate than a diffusion, especially
if one considers a single realization of the physical system of interest. In particular, it
may be strongly inhomogeneous in phase space due to localized objects restricting it:
the cantori described now.
When a KAM torus breaks up, it becomes a Cantor set called a cantorus
(Aubry 1978, Percival 1980) (Pedagogy: section IIB of (Meiss 2015)). In 1984, MacKay,
Meiss, and Percival show that a cantorus is a leaky barrier for chaotic orbits, and that
the flux through the cantorus between two successive iterates of the Poincare map can be
computed as the area of a turnstile built in a way similar to homoclinic lobes for X-points
(MacKay, Meiss & Percival 1984a) (Pedagogy: section IIA of (Meiss 2015)). The above-
described renormalization theories for KAM tori provide a critical exponent for this area
(MacKay et al. 1984a). The latter can be obtained from the actions of homoclinic orbits
(MacKay, Meiss & Percival 1987). A new description of transport in a chaotic domain
can be obtained through Markov models combining the fluxes through the discrete
set of the most important noble cantori (MacKay, Meiss & Percival 1984b, Meiss &
Ott 1985, Meiss & Ott 1986). Such models also enable computing the power law
temporal decay of correlations and lifetimes (Hanson, Cary & Meiss 1985, Meiss &
Ott 1986) first noted in (Chirikov & Shepelyansky 1981, Karney 1983). All these
ideas turn out to be very useful in the next decades, as explained in (Meiss 2015).
An approach to barriers in a chaotic domain motivated by plasma physics consists in
defining approximately invariant circles† (Dewar & Meiss 1992).
The Lyapunov exponent measures the mean rate of divergence of nearby orbits
inside a chaotic domain. It gives a rough estimate of the decay rate of the exponential
part of the correlation functions, which is important in several plasma physics problems
(Grebogi & Kaufman 1981). It is generally computed numerically, but analytical
estimates are available for some mappings (see (Rechester, Rosenbluth & White 1979),
and sections 5.2 and 6.3 of (Chirikov 1979) where it is called Krylov-Kolmogorov-Sinai
entropy). It can be analytically computed for the motion of a particle in a broad
spectrum of waves with a large amplitude (see section 6.8.2 of (Elskens & Escande 2003)).
† This approach leads in 2008 to the definition of temperature contours for chaotic magnetic fields
(Hudson & Breslau 2008). This work quotes a series of studies by non plasma physicists which led to
it, with the introduction of ghost surfaces in particular.
Contributions of plasma physics to chaos and nonlinear dynamics 15
2.3.6. Symbolic dynamics for chaotic layers A striking regularity is present in the
time series of the long chaotic orbits of the standard map that are in a stochastic layer
surrounding a single island and bounded by two KAM tori: the radial coordinate of
the moving point oscillates for a certain time in a region adjacent to an island chain,
then jumps suddenly to another basin, where it remains for a random time, etc... This
behavior can be adequately modelled by a Continuous Time Random Walk (CTRW)
(Balescu 1997). The issue of the diffusion of magnetic field lines in a tokamak leads to
reconsider it in (Misguich, Reuss, Elskens & Balescu 1985). The associated time series
can be described in terms of an algorithm based on a symbolic dynamics. A computer
program enables a completely automatic measurement of the waiting times and of the
transition probabilities of the CTRW, and therefore the analysis of arbitrary long time
series.
2.3.7. Transport in low shear or shearless systems As explained at the end of section
2.2.2, the study of magnetic chaos in systems with reversed shear configurations
motivated the introduction of the standard non-twist map. On varying the control
parameter of this map above the break-up of the shearless curve, it is found that
transport develops very slowly, because of structures with high stickiness giving rise to
an effective barrier near the broken shearless curve (Szezech, Caldas, Lopes, Morrison
& Viana 2012).
Internal transport barriers in toroidal pinches (tokamak and reversed field pinch)
are favored by low or vanishing magnetic shear (del Castillo-Negrete & Morrison 1992b).
This leads Firpo into the study of corresponding Hamiltonian models for the magnetic
field lines, which brings conclusions with a general bearing. Indeed, low shear is shown
to have a dual impact: away from resonances, it induces a drastic enhancement of the
resilience to chaotic perturbations and decreases chaotic transport; close to low-order
rationals, the opposite occurs (Firpo & Constantinescu 2011).
3. Quasi order and order
3.1. Adiabatic theory
When dealing with configurations for the magnetic confinement of charged particles,
one often finds that the motion of a particle in such a configuration has multiple scales.
For instance, section 2.2.1 considered the case of magnetic mirror traps where a particle
has a fast Larmor rotation and slow oscillations along the lines of force. If the dynamics
is not in a regime of large scale chaos, it is natural to take advantage of the time scale
separation to describe the motion. This leads to tractable analytical calculations if
the fast degree of freedom is nearly periodic compared to the slow one: one makes a
(classical) adiabatic theory of the motion. In reality, the adiabatic ideas carry over to
some non strictly adiabatic cases: this is neo-adiabatic theory. The applications of these
ideas are now described.
Contributions of plasma physics to chaos and nonlinear dynamics 16
3.1.1. Classical adiabatic theory Classical adiabatic theory is formalized in 1937
(Krylov & Bogolyubov 1937). In the 1950s and early 1960s, Kruskal is working on
asymptotics and on the preservation or destruction of magnetic flux surfaces. His
unpublished work motivates Lenard and Gardner to develop a theory of adiabatic
invariance to all orders (Lenard 1959, Gardner 1959). He then develops a Hamiltonian
version of adiabatic theory (Kruskal 1962) where adiabatic invariants are related to
proper action variables. This technique is used to second order in (Northrop, Liu &
Kruskal 1966, McNamara & Whiteman 1967). However its implementation is tedious,
which prompts the use of other techniques: first the Poisson bracket technique with a
methodological contribution from plasma physicists (McNamara & Whiteman 1967),
and then the powerful Lie transform method with two important methodological
contributions from plasma physicists in 1976: one by Dewar (Dewar 1976), and one
by Dragt and Finn (Dragt & Finn 1976)† (Pedagogy: section 2.3 of (Lichtenberg &
Lieberman 2013)). Adiabatic motion in plasma physics is also a source of inspiration
for pure mathematicians, as can be seen in (Arnold 1963b) which deals, in particular,
with magnetic traps, and quotes Kruskal’s work in his section devoted to adiabatic
invariants.
3.1.2. Neo-adiabatic theory Several problems in plasma physics where there is a slow
variation of the system of interest cannot be addressed by classical adiabatic theory.
This is in particular the case when this slow variation induces a transition from trapped
to passing orbits in magnetic configurations of magnetic fusion or of the magnetosphere.
Then orbits cross a separatrix. Since the period of a motion diverges on a separatrix,
whatever slow be the evolution of the mechanical system, classical adiabatic theory
breaks down to describe this crossing. However, it turns out that one can still take
advantage of a separation of time scales for most crossing orbits: those which do not stick
too long to the X-point. In 1986, four (groups of) authors come up with the calculation
of the change of adiabatic invariant due to separatrix crossing: (Neishtadt 1986),
(Hannay 1986), (Vasilev & Guzev 1986), and a group of plasma physicists (Tennyson,
Cary & Escande 1986, Cary, Escande & Tennyson 1986). The approaches are very
similar (except for the third paper) and constitute what is now called neo-adiabatic
theory‡. The theory provides also explicit formulas for the trapping probabilities in a
resonance region.
3.1.3. Adiabatic description of Hamiltonian chaos Section 2.3.1 considered the case of
diffusive transport of a particle in strongly overlapping longitudinal waves. The diffusive
picture was justified by the locality in velocity of the wave-particle interaction. This
locality is quantified by a width in velocity which grows with the overlap parameter.
† Cary’s tutorial paper (Cary 1981a) provides a unifying view on Lie transform perturbation theory
for Hamiltonian systems together with important applications in plasma physics.‡ (Bazzani, Frye, Giovannozzi & Hernalsteens 2014) provides an extensive list of papers on neo-
adiabatic theory. An early work already gave the solution in the case of a pendulum (Timofeev 1978).
Contributions of plasma physics to chaos and nonlinear dynamics 17
Then, the diffusive picture is justified if this width is much smaller than the range of
the phase velocities of the waves with strong resonance overlap. In the opposite case,
the locality in velocity of the wave-particle interaction corresponds to a motion where
the trapping time in the frozen potential of all waves is much smaller than the time
scale of variation of this potential (this corresponds to the case of a large Kubo number
introduced in section 2.3.1). At a given time, the frozen potential displays one or more
separatrices which are pulsating with time. This issue is of interest to plasma physicists†.
The simplest case corresponds to a single pulsating separatrix, as occurs for the
dynamics of a nonlinear pendulum in a slowly modulated gravity field. Numerical
simulations reveal that the domain swept by the slowly pulsating separatrix in the
Poincare map looks like a chaotic sea where no island is visible (Menyuk 1985, Elskens
& Escande 1993) (Pedagogy: section 1 of (Elskens & Escande 1993) and section 5.5.2
of (Elskens & Escande 2003)). As a result one might think the limit of infinite overlap
to correspond to some “pure” chaos. A fact pushing in this direction is a theorem
stating that, in the domain swept by the slowly pulsating separatrix, the homoclinic
tangle is tight‡ (Elskens & Escande 1991). However another theorem tells the total
area covered by small islands in the same domain generally decreases when the slowness
of the system increases, but remains finite for symmetric frozen potentials (Neishtadt,
Sidorenko & Treschev 1997). This shows that, when taking at random initial conditions
in the apparently smooth chaotic sea of the motion of a nonlinear pendulum in a slowly
modulated gravity field, there is a finite probability to find a regular orbit: chaotic
does not mean stochastic§! This also shows that chaos is not pure at all, and that the
numerical simulation of orbits may provide a misleading information‖. In the adiabatic
limit, successive separatrix crossings are not independent, which significantly affects
transport (Bruhwiler & Cary 1989, Cary & Skodje 1989). However the separation of
nearby orbits is intuitive, since two such orbits may be separated when coming close
to the X-point, one staying untrapped and the other one becoming trapped. The
transition from stochastic diffusion in a large set of waves to slow chaos associated to a
pulsating separatrix can be detected experimentally in a traveling wave tube (Doveil &
† In 1997, the understanding of adiabatic chaos leads to finding a way of mitigating its effects, such
as in the work on omnigenous stellarators, viz. stellarators where all orbits are confined (Cary &
Shasharina 1997).‡ When resonance overlap diminishes, at some moment the heteroclinic intersection between manifolds
of the two resonances vanishes. This occurs at a threshold approximately given by the resonance
overlap criterion if the two resonances are not too different in size and wavelength (Escande &
Doveil 1981b, Escande 1985).§ Furthermore, the presence of the small islands induces the apparent intermittent trapping of chaotic
orbits in a way analogous to what described in the first footnote of paragraph 2.2.1.‖ If the mathematical model is thought as the approximation of a true physical system, the dynamics
of the latter undergoes actually perturbations like noise. These perturbations are likely to smear out
the many minuscule islands of (Neishtadt et al. 1997). Then, the above numerical simulation gives
the right physical picture. This sets the important issue of the structural stability of mathematical
models when embedded into more realistic ones: numerical simulations might be more realistic than
the mathematical model they approximate!
Contributions of plasma physics to chaos and nonlinear dynamics 18
Macor 2011).
Adiabatic invariants of slowly varying Hamiltonian systems occur not only for
almost periodic orbits, but also for chaotic orbits that wander ergodically over the energy
surface of the system. This type of invariant is important in statistical mechanics of
many-body systems, but is also invoked by Lovelace (Lovelace 1979) for single particle
dynamics in the context of beam-plasma equilibrium and stability. This leads Ott to
consider the general question of how well these approximate constants are preserved.
Using multiple time scale techniques and adopting ideas from quasilinear theory, he
shows, among other results, that the error in these invariants is much larger than the
one for almost periodic orbits (Ott 1979).
3.1.4. Separatrix crossing for non-slowly varying dynamics In 2015, Benisti shows
that for the rapidly varying dynamics of a particle in a sinusoidal wave with a large
exponential growth and a small initial amplitude, one can still describe separatrix
crossing (Benisti & Gremillet 2015). This is done in two steps. First, a perturbative
analysis in the wave amplitude provides the change in action up to the time when
the action remains nearly stationary after trapping. Then, adiabatic theory is used
to describe the subsequent evolution of the orbit: the perturbative and adiabatic
descriptions are matched. The method can be generalized to non-sinusoidal potentials
and to waves that do not simply grow exponentially in time†. This technique works
for particles with a high enough initial velocity. Lower velocities can be described by
neo-adiabatic theory (Benisti & Gremillet 2015).
3.2. Chimeras
Fast ions in a fusion reactor can excite many types of waves, and in particular the
energetic particle mode (EPM) which can produce avalanches of such ions. The coherent
nonlinear behaviour of an EPM can be described by the complex Ginzburg-Landau
equation (equation (2) of (Zonca, Briguglio, Chen, Fogaccia, Hahm, Milovanov &
Vlad 2006)), which describes a vast array of phenomena including nonlinear waves,
second-order phase transitions, Rayleigh-Benard convection and superconductivity. In
2014, Sethia and Sen find out that this equation gives chimera states (Sethia & Sen 2014).
These states represent a spontaneous breakup of a population of identical oscillators that
are identically coupled, into subpopulations displaying synchronized and desynchronized
behavior. Till then, they had been found to exist in weakly coupled systems and with
some form of nonlocal coupling between the oscillators. The new result shows that
neither of these conditions is essential for their existence.
† This method can also derive the particles’ distribution function. This allows to easily calculate the
nonlinear response of a cold beam to an electrostatic wave (Benisti & Gremillet 2015).
Contributions of plasma physics to chaos and nonlinear dynamics 19
3.3. Solitons and solitary waves
Year 1965 brings a landmark in soliton† theory: the discovery by Zabusky and Kruskal
(Zabusky & Kruskal 1965) that Korteweg-de Vries‡ (KdV) solitary waves survive
collision. At that time computers enable to compute the collision of two solitary waves
(their existence was proved in 1895). The outcome is surprising: after the collision, both
solitary waves recover their initial shapes. Two years after, the solution is provided by
Gardner, Greene, Kruskal, and Miura, with the inverse scattering transform§ (Gardner,
Greene, Kruskal & Miura 1967, Gardner, Greene, Kruskal & Miura 1974) which is a key
technique of soliton theory with a lot of applications beyond plasma physics. During
the period 1970 to 1982 many experiments on solitons in plasmas were performed (see
(Lonngren 1983) for a review).
In 1970, Kadomtsev and Petviashvili (Kadomtsev & Petviashvili 1970) introduce
the first partial differential equation generating solitons in two space dimensions, as a
model to study the evolution of long ion-acoustic waves of small amplitude propagating
in plasmas under the effect of long transverse perturbations. This equation is a universal
model (a kind of normal form) for nonlinear, weakly dispersive waves in an anisotropic
medium (Biondini & Pelinovsky 2008).
In 1971, Rogister shows that the evolution of small-amplitude nonlinear Alfven
waves propagating quasiparallel with respect to the background magnetic field is
governed by an equation called the derivative nonlinear Schrodinger equation (Rogister
1971). In 1978, Kaup and Newell prove that this equation has solitonic solutions too
(Kaup & Newell 1978).
The effect of energy dissipation on solitary waves is important for plasma physics.
This starts to be examined in 1969 by Ott and Sudan. Using a multiple time scale
technique, they analyze the damping of solitary waves, first for the case of ion acoustic
waves (Ott & Sudan 1969), and subsequently for very general types of damping
mechanisms (Ott & Sudan 1970).
In 1983, Kaw, Sen and Valeo initiate an approach to the evolution of interacting
waves in plasma, which consists in reducing the initial problem stated in terms of partial
differential equations into a nonlinear Hamiltonian form with two degrees of freedom
only (Kaw, Sen & Valeo 1983). This permits the identification and classification of
a rich variety of nonlinear solutions, in particular solitary waves, but also periodic
and stochastic solutions (Kaw et al. 1983, Kaw, Sen & Valeo 1985, Kaw, Sen &
Katsouleas 1992, Bisai, Sen & Jain 1996). Reference (Kaw et al. 1983) elicits also
interest in the nonlinear dynamics community because of the curious square root form
of the nonlinear potential which seemed to make the system fully integrable. Eventually
the system was shown to be non-integrable by an application of Ziglin’s theorem
† There is a wealth of textbooks on solitons; the Wikipedia article may be a good pedagogical
startpoint.‡ The Korteweg-de Vries equation describes long-wavelength water waves and ion-acoustic waves in
plasmas.§ Pedagogy: the Wikipedia entry with this title.
Contributions of plasma physics to chaos and nonlinear dynamics 20
(Grammaticos, Ramani & Yoshida 1987).
In 2012, Sen’s research work on solitons in plasmas brings an interesting spin-off.
A genetic programming search for equations sharing the KdV solitary wave solution
uncovers a KdV-like advection-dispersion equation and an infinite dimensional family
of equations with this property (Sen, Ahalpara, Thyagaraja & Krishnaswami 2012).
4. High dimensional Hamiltonian dynamics
4.1. Self-consistent dynamics
4.1.1. Finite dimension Self-consistent dynamics is at the heart of the complex
behavior exhibited by plasmas. Indeed, the electric and magnetic fields determining
the motion of charged particles are generated by the particles themselves. As a result,
the particle dynamics in a plasma corresponds to a coupled Hamiltonian dynamical
system with a very large number of degrees of freedom. This occurs in particular in the
wave-particle interaction. References (Escande 1991, Antoni, Elskens & Escande 1998)
and chapter 2 of (Elskens & Escande 2003) derive the Hamiltonian describing the self-
consistent evolution of N ′ tail particles and M longitudinal waves, by starting from
the N -body description of a one-dimensional plasma with a finite length (Pedagogy:
section 2.1 of (Elskens & Escande 2003)). As explained in section 2.3.1, this derivation
is motivated by a finite dimensional approach to the explanation of the surprising
experimental result of (Tsunoda et al. 1987a, Tsunoda et al. 1987b, Tsunoda et al. 1991).
The case with many waves displays a phenomenon known in fluid dynamics as
“depletion of nonlinearity”: if the tail distribution function is a plateau in both velocity
and space, which occurs at the saturation of the bump-on-tail instability introduced in
section 2.3.1, the self-consistency is quenched, since the particles are not able to modify
the wave amplitudes (see section 2.2 of (Besse, Elskens, Escande & Bertrand 2011)).
Therefore, their dynamics, even when strongly chaotic, is the 1.5 degree-of-freedom
one corresponding to the motion in a prescribed spectrum of waves. Depending on
this spectrum, the diffusion coefficient may be quasilinear or not, as recalled in section
2.3.1. This contradicts previous works trying to prove the validity of quasilinear theory
(Liang & Diamond 1993b, Liang & Diamond 1993a, Escande & Elskens 2002b, Elskens
& Escande 2003) and the “turbulent trapping” Ansatz aiming at the contrary (Laval &
Pesme 1984) (see section 2.3 of (Besse et al. 2011)). This brings a first element toward
understanding the surprising experimental result recalled above.
The derivation for the single wave case (M = 1) was first provided in two
seminal papers (Onishchenko, Linetskiı, Matsiborko, Shapiro & Shevchenko 1970,
O’Neil, Winfrey & Malmberg 1971). It was also provided in (Tennyson, Meiss &
Morrison 1994, del Castillo-Negrete 1998, Crawford & Jayaraman 1999) by starting
with kinetic descriptions. The single wave case is a paradigm for the interaction of
particles with collective degrees of freedom: electrostatic instabilities (Crawford &
Jayaraman 1999), interaction of vortices with finite-velocity flow in hydrodynamics (del
Contributions of plasma physics to chaos and nonlinear dynamics 21
Castillo-Negrete 2000, del Castillo-Negrete & Firpo 2002, del Castillo-Negrete 2005).
These references study the corresponding transition to chaos and formation of coherent
structures in phase space: order and chaos coexist, a typical feature of complex systems,
already indicated by the existence of chimeras. The M = 1 case also brings the
possibility of analytical calculations which are impossible in the full N -body description
of the plasma. Indeed, Gibbs statistical mechanics can be derived in this case, revealing
a second order phase transition associated with the Landau damping regime: for
nonequilibrium initial data with warm particles, above a critical initial wave intensity,
thermodynamics predicts a finite wave amplitude in the limit N → ∞; below it, the
equilibrium amplitude vanishes (Firpo & Elskens 2000).
4.1.2. Infinite dimension Due to its interaction with resonant particles, a longitudinal
wave in a thermal plasma experiences a non-dissipative damping discovered by Lev
Landau in 1946. This effect is of paramount importance in plasma physics. In a
Vlasovian approach, it is understood as the consequence of a phase-mixing effect of
a continuum of linear modes, called van Kampen modes†. However, one may wonder
whether nonlinear effects do not destroy these linear modes and the corresponding phase
mixing. Proving the innocuity of nonlinear effects is the equivalent of deriving a KAM
theorem for a continuous system (the Vlasov-Poisson one), a tour de force which partly
earned Villani the 2010 Fields medal (Mouhot & Villani 2010, Villani 2014). This is a
major contribution of plasma physics to the nonlinear dynamics of continuous systems‡.
When the electron distribution function is gradually changed, for instance by adding
a bump in the tail of the distribution, an instability may appear in the Vlasov-Poisson
system describing a spatially uniform plasma. For such a problem, center manifold
theory cannot be used because of the existence of a continuum of modes. In order to cope
with this problem, in 1989, Crawford and Hislop use the method of spectral deformation,
a technique till then used in the theory of Schrodinger operators in quantum mechanics.
They derive equations for the nonlinear evolution of electrostatic waves by extending the
method to the full nonlinear Vlasov equation, without making the standard assumptions
of weak nonlinearity and separated time scales (Crawford & Hislop 1989). In 1994,
Crawford overcomes the absence of a finite-dimensional center manifold in this problem,
by restricting his analysis to initial conditions where only the unstable mode is initially
excited, so that it is not one component of an arbitrary fluctuation. This way, he can
treat the Vlasov equation perturbatively, and shows that for a plasma with a neutralizing
background of ions, the instability saturates at an amplitude scaling like the square of
† (Morrison 2000a) solves the dynamics of this continuum in the context of Hamiltonian systems theory,
by a canonical transformation to action-angle variables for this infinite degree-of-freedom system.‡ The physical interpretation of Landau damping is subtle, but can be made completely intuitive
by using the above finite dimensional self-consistent approach which shows that it is due to the
synchronization of quasi-resonant particles with the wave (Escande, Zekri & Elskens 1996, Elskens
& Escande 2003); the synchronization is confirmed experimentally (Doveil, Escande & Macor 2005).
Within this approach, proving Landau damping in a nonlinear context requires the standard KAM
theorem only.
Contributions of plasma physics to chaos and nonlinear dynamics 22
the growth rate, in heavy contrast with the traditional scaling like the square root.
Furthermore, all orders contribute to the saturation, which implies that the equilibrium
may be approached in a non monotonic way (Crawford 1994).
As already indicated in section 3.2, the self-consistent interaction of fast ions with
waves is an important topic for burning magnetic fusion plasmas. An analogy mapping
the previous velocity distribution to the radial distribution of particles in a toroidal
plasma provides a description of the corresponding limit case of a uniform plasma by
the continuous limit of the above finite dimensional self-consistent model (Zonca, Chen,
Briguglio, Fogaccia, Vlad & Wang 2015). In 2015, Zonca and coworkers show that
the more general description of the magnetic fusion case is provided by an analogue of
Dyson’s equation in quantum field theory, describing particle transport due to emission
and reabsorption of toroidal symmetry breaking perturbations, called “phase space
zonal structures” (PSZS), by analogy with the meso-scale configuration space patterns
spontaneously generated by drift-wave turbulence. The relevant dynamics corresponds
to the non-adiabatic (chaotic) case where the particle trapping time is not short with
respect to the characteristic time for the nonlinear evolution of PSZS. For a non uniform
plasma, there is a convective amplification of wave packets as avalanches leading to the
secular transport of particles over large radial scales inside the toroidal plasma. This
physics has analogies with the “super-radiance” regime in free-electron lasers (Zonca
et al. 2015).
Avalanches are also present in gyrokinetic numerical simulations of tokamak
plasmas, which correspond to a reduced Vlasovian description of such plasmas taking
advantage of the fact that the magnetic moment of particles is an adiabatic invariant.
Avalanches correspond to a description of transport at strong variance with that
of a chaotic transport due to turbulent waves. This description is germane with
self-organized criticality, the kind of self-organization at work in sandpiles. This
view progressively emerged in the last two decades (Dendy & Helander 1997, Dendy,
Chapman & Paczuski 2007, Sanchez & Newman 2015).
4.2. Noncanonical Hamiltonian theory
The standard Hamiltonian description of physical systems uses canonical variables. This
makes this description uneasy for systems that are written in terms of noncanonical
variables like ideal fluid and systems with long range interactions when described by
Vlasov equation. This motivates Greene and Morrison to introduce in 1980 noncanonical
Poisson brackets for fluid systems (Morrison & Greene 1980). Morrison brings this
approach to full maturity in the following two decades (see (Morrison 1998) for a review).
This approach leads in particular to the energy-Casimir criterion of nonlinear stability†
(see section VI.B of (Morrison 1998)). Furthermore, linearly stable equilibria with
negative energy modes are shown to be unstable when nonlinearity or dissipation is
† This criterion was introduced first in (Holm, Marsden, Ratiu & Weinstein 1985), and in a series of
papers where several plasma physicists are present (see note 42 of (Morrison 1998)).
Contributions of plasma physics to chaos and nonlinear dynamics 23
added (see section VI of (Morrison 1998) for a global discussion).
5. Dissipative dynamics
While a wealth of plasma physics issues are naturally dealt with by a Hamiltonian
approach, dissipative effects are important in plasmas too†. This is the case for
nonlinear wave coupling. In particular, for the dynamics of an unstable wave coupled
nonlinearly to two lower frequency damped waves (Vyshkind & Rabinovich 1976). In
1980, Wersinger, Finn, and Ott demonstrate via the Poincare section technique that
this dynamics is well described by a one-dimensional map with a quadratic maximum‡,
which implies chaos with a strange attractor (Wersinger, Finn & Ott 1980). This is the
first explicit numerical demonstration of the applicability of such a map from physically
motivated differential equations. As a sequel of this work, numerical techniques are
used to compute for the first time the fractal dimension of strange attractors in
several examples. These dimensions are then compared with the predictions from the
Kaplan-Yorke conjecture, which relates an attractor’s fractal dimension to its Lyapunov
exponents, thus providing important early confirmation for the conjecture (Russell,
Hanson & Ott 1980).
In 1982, Bussac publishes an analytical method which accounts for the main
features of the asymptotic solution of this coupled wave dynamics for the nonlinear
decay of a coherent unstable wave into its subharmonic (Bussac 1982b). This dynamics
is shown to be close to a Hamiltonian one. The map providing the mismatch to
the corresponding constant energy at each crossing of the Poincare surface of section
exhibits period doubling, and an explicit equation is obtained for the chaotic attractor.
The same type of nonlinear wave coupling can lead to another path to chaos called
type I intermittency. When the control parameter is varied, the transition to chaos is
abrupt, but the quantities which measure the amount of chaos smoothly vary with the
control parameter, which implies a continuous character for such a transition (Bussac &
Meunier 1982c). Reference (Bussac 1982a) yields a complete panorama of the dynamics
of the three-wave system involving again a one dimensional map which is analytically
derivable from the original system of differential equations.
In 2003, Firpo shows that the description of the early nonlinear regime of the
resistive m = n = 1 mode of the tokamak can be done by assuming that the perturbation
retains the form of the linearly unstable eigenmode, which leads to a Landau nonlinear
stability equation (Firpo & Coppi 2003, Firpo 2004, Firpo 2005). The difficulty and
novelty of this analysis comes from the existence of an inner critical layer whose position
is not fixed, in contrast with those occurring next to walls in fluid (Stuart 1958) and
† See (Greiner, Klinger, Klostermann & Piel 1993, Klinger, Greiner, Latten, Piel, Pierre, Bonhomme,
Arnas-Capeau, Bachet & Doveil 1995, Mausbach, Klinger & Piel 1999) for a simple experimental proof
of various phenomena like period doubling bifurcations and intermittency in low-pressure thermionic
discharges. Pedagogy: chapter 7 of (Lichtenberg & Lieberman 2013).‡ This type of map exhibits period doubling cascades.
Contributions of plasma physics to chaos and nonlinear dynamics 24
other plasma physics problems (Dahlburg 1998) whose solutions follow similar paths. In
the much simpler problem of the nonlinear tearing mode where the position of the inner
critical layer is fixed, Ottaviani and the author of the present paper show in 2004 that the
perturbation can be written as a linear unstable eigenmode plus a higher-order correction
ψ whose radial width scales like the linear magnetic island width. This yields a linear
differential equation for ψ, which can be solved readily (Escande & Ottavani 2004).
6. Quantum chaos
In 1979, the issue of the chaotic motion of geometric optics rays in plasmas motivates
McDonald and Kaufman to study the quantum chaos provided by the two-dimensional
Helmholtz equation with “stadium” boundary. In contrast to the clustering found for a
separable equation, the eigenvalue separations have a Wigner distribution, characteristic
of a random Hamiltonian (McDonald & Kaufman 1979).
The same year, a work involving Chirikov considers the quantum kicked rotator,
which is the quantized version of the standard map. In the regime of strong chaos,
the rotator energy and the squared number of excited quantum levels appear to grow
diffusively in time, as in the corresponding classical system. Only up to a break time
though (Casati, Chirikov, Ford & Izrailev 1979), after which the quantum energy
excitation is suppressed while the classical one continues to diffuse. This time is
proportional to the classical diffusion rate (Chirikov, Izrailev & Shepelyansky 1981).
However, even classically small noise can induce quantum decoherence, thus
eliminating the quantum saturation of diffusion, and restoring perpetual classical
diffusion (Ott, Jr & Hanson 1984). This type of behavior has been experimentally
investigated in atomic physics experiments (Arndt, Buchleitner, Mantegna & Walther
1991).
Multi-photon ionization of hydrogen can be treated by classical theory when the
initial quantum number is large and the photon energy is small. Then ionization
corresponds to the chaotic motion obtained for high photon intensities. However, the
quantum ionization threshold may be higher than the threshold of classical chaos,
because classical chaos is suppressed by quantum effects when the phase-space area
escaping through classical cantori each period of the electric field is smaller than Planck’s
constant (MacKay & Meiss 1988b).
7. Natural extensions
There are natural extensions to this work. In particular, those due to intertwined
contributions of plasma physicists with fluid dynamicists about vortex structures (see
chapter 6 of (Horton & Ichikawa 2002)). Also about the generalized Lagrangian
mean, a formalism to unambiguously split a motion into a mean part and a nonlinear
oscillatory part, where plasma physicists provided important contributions (Frieman
& Rotenberg 1960, Dewar 1970). There is also the use of non-neutral plasmas
Contributions of plasma physics to chaos and nonlinear dynamics 25
for the experimental study of vortex dynamics in two-dimensional hydrodynamics
(Malmberg 1992, Durkin & Fajans 2000).
8. Conclusion
The main part of this story is borne out of two constraints: the need for a theoretical
description of the magnetic confinement of thermonuclear plasmas and the insufficient
corresponding mathematical knowledge. However its birth would not have been possible
without the development of computer calculations.
Plasma physics is often considered as motivated by its applications. This is wrong,
since plasmas belong in complex systems which are of interest on their own, and are a
central topic of contemporary physics. This is right, provided one counts applications
like its just recalled numerous contributions to chaos and nonlinear dynamics!
A histogram of the number of published papers quoted in this topical review as a
function of time shows that there is a peak of the number of published papers per year
in the 1980-1984 period, then a decrease to 60% of this peak in the 1985 to 2004 period,
and to 40% in the 2005-2014 period, with no decrease when going from 2005-2009 to
2010-2014. Therefore, there still is a steady flow of contributions from plasma physics
to chaos and nonlinear dynamics.
However, there is less excitement about these new results than in the 1980-
1984 period where plasma physicists started to understand chaos. The 1980-1984
results recalled in this topical review bring a reassuring background to the present
investigations, but most are not much used explicitly. Indeed, as to fusion theory, the
development of massive computer simulations progressively revealed that the complexity
of fusion plasmas displays a mixture of order and chaos, for instance in avalanches,
that is not suggestive of an elementary quantitative description (so far). Furthermore,
numerical calculations are so handy, that it is more reliable to find out a threshold
of bifurcation numerically, for instance a threshold of chaos, than to compute it by
applying a low dimensional technique after painstaking uncontrolled approximations.
The practical success of the simple-minded quasilinear approximation was another
incentive to give up more sophisticated descriptions of chaotic transport. Works about
nonlinear or chaotic plasma physics have been taking over: a whole series of books would
be necessary to summarize their results! However a breakthrough in the fundamental
analysis of complex plasma dynamics cannot be excluded...
I thank P. Diamond, U. Frisch, and Y. Pomeau for inciting me to study the topic of
this topical review. I thank S. Abdullaev, D. Benisti, J. Cary, D. del-Castillo-Negrete,
R. Dewar, F. Doveil, Y. Elskens, M.-C. Firpo, U. Frisch, J. Krommes, R. MacKay, J.
Meiss, J.T. Mendonca, J. Misguich, P. Morrison, E. Ott, A. Sen, D. Shepelyansky, J.
B. Taylor, M. Vlad, and F. Zonca for their initial inputs to this work and for successive
suggestions. Also, Y. Elskens, R. MacKay, and P. Morrison who provided me extensive
advice on my manuscript. I thank R. Dendy for suggestions and also for questions, which
fed the concluding remarks, and L. Couedel for his help in managing the references.
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